I saw MacCavity's solution. While that is technically correct- it doesn't give much confidence when you have a subset of only one and a set of two
That is correct. The calculation demonstrates the concept of conditional probability, but the sample size is too small to hold any significance.
Maybe one could estimate the probability by taking his pessimistic 50% chance and combine with the more optimistic 10% chance. How would you do that? As an average? That would give 30%. Sounds reasonable.
It may "sound reasonable", but it is not. The 10% calculation is based on a population of samples that are outside of the set that are relevant to the problem. The 50% solution is, as you have described it above, too small to be meaningful. What you are left with is no meaningful calculation to predict the end of this cycle based on the duration of past cycles.
The life expectancy table you posted is interesting, and might be useful for illustration, but I don't know what the "Life Expectancy Factor" means in terms of probability. I can guess that it's the average number of years of remaining life for any age group, but knowing the average does not tell you anything about the distribution. One should be able to work back from the table to figure that out (if the interpretation is correct), but I'm not inclined to spend the time doing that.
Perhaps the point can be reinforced by considering a common misunderstanding of probability on the flip side of this conditional probability problem. A fair coin has a 50% chance of coming up heads or tails, which can be verified by empirical data, and this probability is UNconditional. If you flip a coin enough times, the number of heads and tails will approach even numbers. So if you start out flipping a fair coin and you get 10 heads in a row, what is the probability of getting another head? In this case, past history is again irrelevant because their is no "condition" that applies. Each flip has a 50/50 chance of coming up heads or tails, regardless of what has happened on all previous flips. This is very different from a game of Black Jack, where the probability of cards being drawn from the remaining deck depends very much upon which cards have already been removed. This is why "card counting" a la Rain Man is frowned upon by casinos, and use of devices to assist in counting is strictly prohibited. |
